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Constraint Singularity-Free Design of the IRSBot-2
Coralie Germain, Sébastien Briot, Stéphane Caro, Philippe Wenger
To cite this version:
Coralie Germain, Sébastien Briot, Stéphane Caro, Philippe Wenger. Constraint Singularity-Free
De-sign of the IRSBot-2. Advances in Robot Kinematics, Jun 2012, Innsbruck, Austria. �hal-00676851�
IRSBot-2
Coralie Germain, S´ebastien Briot, St´ephane Caro and Philippe Wenger
Abstract This paper deals with the constraint analysis of a novel two-degree-of-freedom (DOF) spatial translational parallel robot for high-speed applications named the IRSBot-2 (acronym for IRCCyN Spatial Robot with 2 DOF). Unlike most two-DOF robots dedicated to planar translational motions this robot has two spatial kinematic chains that provide a very good intrinsic stiffness. First, the robot architecture is presented and its constraint singularity conditions are given. Then, its constraint singularities are analyzed in its parameter space based on a cylindrical algebraic decomposition. Finally, a deep analysis is carried out in order to determine the sets of design parameters of the IRSBot-2 that prevent it from reaching any con-straint singularity. To the best of our knowledge, such an analysis is performed for the first time.
Key words: parallel manipulator, constraint singularity, cylindrical algebraic de-composition, design.
1 Introduction
Several robot architectures with two translational degrees of freedom (DOF) for high-speed operations have been proposed in the past decades. Brog˚ardh proposed in [2] an architecture made of a parallelogram joint (also called Π joint) located between the linear actuators and the platform. Another two-DOF translational robot was presented in [5], where the authors use two Π joints to link the platform with two vertical prismatic actuators. Its equivalent architecture actuated by revolute joints is presented in [4].
Coralie Germain, S´ebastien Briot, St´ephane Caro, Philippe Wenger
Institut de Recherche en Communications et Cybern´etique de Nantes, France e-mail:
{germain,briot,caro,wenger}@irccyn.ec-nantes.fr
2 Coralie Germain, S´ebastien Briot, St´ephane Caro and Philippe Wenger
The foregoing architectures are all planar, i.e., their elements are constrained to move in the plane of motion. As a result, their elements are all subject to bending effects in the direction normal to the plane of motion. In order to guarantee a mini-mum stiffness in this direction, the elements have to be bulky, leading to high inertia and low acceleration capacities. In order to overcome these problems, a new Delta-like robot, named the Par2, was proposed in [7]. However, even if its acceleration capacities are impressive, its accuracy is poor.
A two-DOF spatial translational robot, named IRSBot-2, was introduced in [3] to overcome its counterparts in terms of mass in motion, stiffness and workspace size. The IRSBot-2 has a spatial architecture and the distal parts of its legs are subject only to traction/compression/torsion. As a result, its stiffness is increased and its total mass can be reduced. Nevertheless, the IRSBot-2 may reach some constraint singularities [1, 8]. In this paper, a deep analysis is carried out in order to determine the sets of design parameters of the IRSBot-2 that prevent it from reaching any constraint singularity.
This paper is organized as follows. First, the robot architecture is described and its constraint singularity conditions are given. Then, its constraint singularities are analyzed in its parameter space based on a cylindrical algebraic decomposition. Fi-nally, the set of design parameters for the robot to be free of constraint singularity are determined.
2 Robot Architecture and Constraint Singularity Conditions
The IRSBot-2 is shown in Fig. 1 and is composed of two identical legs linking the fixed base to the moving platform. Each leg contains a proximal module and a distal module, which are illustrated in Fig. 2.
Base x0 y0 z0 Platform Parallelogram Elbow
Fig. 1 CAD Modeling of the IRSBot-2
x0 x0 y0 y0 y0 y0 z0 z0 O Base Platform P0 P1 P2 αi αi Proximal module Distal module
x0 x 0 x0 x0 x0 y0 y0 y0 z0 z0 F2i Ei E1i E2i Fi F1i Hhi Hbi Base Platform P0 P1 P2 α b d l1 a1 a2 p l2 l2eq qi ψi λi β1i β1i β2i β2i e θi
Fig. 3 Paramaterization of the ith leg (i = 1, 2)
` Pi Hhi Fi Hbi Ei l2eq ψi θi a1sin β a2sin β λi Fig. 4 Closed-loop Ei–Hbi–
Hhi–Fi: projection of the
dis-tal module on the plane (x0Oz0)
The parameters of the IRSBot-2 used throughout this paper are depicted in Figs. 3 and 4. From [3], the IRSBot-2 reaches a constraint singularity iff1:
θ1= θ2+ kπ, k= 0, 1 (1)
and
(xP2− xP1) cos 2
β cos θ2− (zP2− zP1) sin θ2= 0 (2)
It is noteworthy that Eqs. (1) and (2) depend only on the design parameters asso-ciated with the distal module. Therefore, the proximal modules of the IRSBot-2 do not affect its constraint singularities and we focus only on the constraint singularities associated with the distal modules.
3 Constraint Singularity Analysis of the IRSBot-2 in its
Parameter Space
This section aims to find the sets of design parameters (a1, a2, β , p, l2eq) that allow
the IRSBot-2 to reach some constraint singularities. Note that the foregoing five de-sign parameters are shown in Fig. 3. a1, a2and l2eqare the lengths of segments EiE1i,
FiF1iand HbiHhi, respectively. p is the moving-platform radius. The coordinates of
vector−−→P1P2can be expressed as:
xP2− xP1 = 2p + ` (cos ψ2− cosψ1) (3) zP2− zP1 = ` (sin ψ2− sinψ1) (4) ` = a2l2eq a1− a2 (5) 1Let β denote β 22, then β11= π + β , β21=−β and β12= π− β
4 Coralie Germain, S´ebastien Briot, St´ephane Caro and Philippe Wenger
Angles ψ1and ψ2are depicted in Figs. 3 and 4. From the closed-loop Ei–Hbi–Hhi–Fi
(i = 1, 2) and Fig. 4, the following relations between λi, θiand ψiare obtained:
l2eqcos ψi= λicos θi− (a1− a2) sin β (6)
−l2eqsin ψi=−λisin θi (7)
λiis depicted in Fig. 4 and is derived from Eqs. (3) to (7):
λi=
q
l22eq+ (a1− a2)2sin2β + 2(−1)i+1l2eqcos ψi(a1− a2) sin β (8)
The following three cases, obtained from Eqs. (1) and (8), allow us to simplify Eqs. (3) to (7) to end up with a univariate polynomial form of constraint singularity condition (2):
Case I: θ1= θ2+ π and λ1= λ26= 0
Case II: θ1= θ2+ π and λ16= λ2
Case III: θ1= θ2
For Case I, Eq. (2) takes the form:
PI(X ) = A1X2+ B1X+C1= 0 (9) with A1 =− l22eqsin2β a2/(a1− a2)
B1 = l2eq(1− sin2β ) ( p− a2sin β )
C1 =− p (a1− a2) (1− sin2β ) sin β + l22eqa2/(a1− a2)
X= cos ψ, ψ = ψ2, X ∈ [−1, 1], [a1, a2, β , p]∈ D,
l2eq∈]0, +∞[
D =]0, +∞[×]0, a1[×[0, π/2]×]0, +∞[.
For Case II, Eq. (2) takes the form:
PII(X ) = A2X2+C2= 0 (10) with A2 = a2sin3β C2 = p(1− sin2β )− a2sin3β X= cos θ , θ = θ2, X ∈ [−1, 0], [a1, a2, β , p]∈ D,
l2eq∈](a1− a2) sin β|sinθ|, (a1− a2) sin β [
For Case III, Eq. (2) takes the form:
PIII(X ) = A3X2+C3= 0 (11)
A3 = a2sin3β C3 = p(1− sin2β )− a2sin3β X= cos θ , θ = θ2, X ∈ [−1, 1], [a1, a2, β , p]∈ D,
l2eq∈](a1− a2) sin β , +∞[
As a matter of fact, the IRSBot-2 reaches a constraint singularity as long as one of the univariate polynomials (9), (10), (11) admits one solution at least. The set of design parameters (a1, a2, β , p, l2eq) for which the constraint singularities
asso-ciated with Cases I, II and III can be reached are obtained with a method based on the notion of Discriminant Varieties and Cylindrical Algebraic Decomposition. This method resorts to Gr¨obner bases for the solutions of systems of equations and is de-scribed in [6]. Besides, the tools used to perform the computations are implemented in a Maple library called Siropa2.
Table 1 Formulae describing the boundaries of the cells in Tables 2, 3 and 4 a11= 0 p1= 0
a12= +∞ p2(a1, a2, β ) =11+sin β−sinβa2sin β
a21= a1 p3(a1, a2, β ) =1−sin 2β 1+sin2βa2sin β a22= +∞ p4(a1, a2, β ) = a2sin β β1= 0 p5(a1, a2, β ) =1+sin 2β 1−sin2βa2sin β β2= arcsin(1/ √
3) p6(a1, a2, β ) =1+sin β1−sinβa2sin β
β3= π/4 p7= +∞ β4= π/2 p8(a1, a2, β ) = a2sin β tan2β l2eq1(a1, a2, β , p) =a1a−a22 p l2eq2(a1, a2, β , p) = (a1− a2) sin β l2eq3(a1, a2, β , p) =2aa1−a2 2sin β q
(sin2β− 1)(sin2β− 1)(p − a2sin β )2+ 4 p a2sin3β l2eq4(a1, a2, β , p) = (a1− a2) sin β|sinθ|
l2eq4(a1, a2, β , p) = +∞
Table 1 provides the different formulae bounding the five-dimensional cells as-sociated with Cases I, II and III. a1and β can be chosen independently. Then, the
boundaries for a2, p are l2eqare determined successively. Table 2 characterizes all
the cells where the IRSBot-2 can reach a constraint singularity, namely, where PI,
PII or PIIIhas at least one real root. It is noteworthy that a real root of one the three
foregoing polynomials amounts to two symmetrical singular configurations of the distal module. It is apparent that six cells arise where PIhas a single real root, two
cells arise where PIhas two real roots. PIIand PIIIcan get two real roots in one cell
only. Some constraint singularities of the IRSBot-2 are shown in3.
2http://www.irccyn.ec-nantes.fr/˜chablat/SIROPA/files/siropa-mpl.html 3http://www.irccyn.ec-nantes.fr/IRSBot2
6 Coralie Germain, S´ebastien Briot, St´ephane Caro and Philippe Wenger
Table 2 Cells of R5where the IRSBot-2 can reach constraint singularities
Case I
(]a11, a12[, ]a21, a22[, ]β1, β4[)
]p1, p2[ (]l2eq1, l2eq2[)
Two singular configs. ]p2, p3[ (]l2eq1, l2eq2[)
]p3, p4[ (]l2eq1, l2eq2[)
]p4, p5[ (]l2eq2, l2eq1[)
]p5, p6[ (]l2eq2, l2eq1[)
]p6, p7[ (]l2eq2, l2eq1[)
]p3, p4[ (]l2eq3, l2eq1[)Four singular configs.
]p4, p5[ (]l2eq3, l2eq2[)
Case II
(]a11, a12[, ]a21, a22[, ]β1, β4[) ]p1, p8[ (]l2eq4, l2eq2[) Four singular configs.
Case III
(]a11, a12[, ]a21, a22[, ]β1, β4[) ]p1, p8[ (]l2eq1, l2eq2[) Four singular configs.
4 Design Parameters for the IRSBot-2 to be Free of Constraint
Singularity
This section aims to find the sets of design parameters (a1, a2, β , p, l2eq) that
pre-vent the IRSBot-2 from reaching any constraint singularity. It amounts to find the intersection of cells where PI, PIIand PIIIdo not have any real root over their mutual
domain.
It turns to be quite difficult to obtain the intersection of cells contrary to their union. As a consequence, we will search for the cells where the product of PI, PII
and PIII does not have any real root. From (10) and (11), it is apparent that the
expressions of PIIand PIII are the same, but their domains are disjointed and
com-plementary because of the bounds of l2eq. Therefore, the sets of design parameters
(a1, a2, β , p, l2eq) that prevent the IRSBot-2 from reaching any constraint singularity
correspond to the union of cells that do not provide any real root for the following two univariate polynomials:
PIV(X ) = PIPII(X ) = (A1X2+ B1X+C1)(A2((X− 1)/2)2+C2) = 0 (12)
with
(
X ∈ [−1, 1], [a1, a2, β , p]∈ D,
l2eq ∈]|sinθ|(a1− a2) sin β , (a1− a2) sin β [
and
PV(X ) = PIPIII(X ) = (A1X2+ B1X+C1)(A3X2+C3) = 0 (13)
with
(
X ∈ [−1, 1], [a1, a2, β , p]∈ D,
l2eq∈](a1− a2) sin β , +∞[
Eq. (12) amounts to the product of PI and PII with a change a variable for PII
and the most restrictive domain for l2eqdefined in (10), whereas Eq. (13) amounts
to the product of PIand PIIIwith the most restrictive domain for l2eqdefined in (11).
Table 1 gives the different formulae bounding the five-dimensional cells associated with (12) and (13). The cells where PIV and PV do not have any real root, i.e., the
sets of design parameters (a1, a2, β , p, l2eq) that prevent the IRSBot-2 from reaching
any constraint singularity, are expressed in Tables 3 and 4, respectively.
Table 3 Cells where Eq. (12) does not have any real root with a1∈]a11, a12[ and a2∈]a21, a22[
[β1, β2[ (]p8, p3[, ]l2eq4, l2eq1[), (]p3, p4[, ]l2eq4, l2eq3[), (]p4, p5[, ]l2eq4, l2eq3[), (]p5, p7[, ]l2eq4, l2eq2[)
[β2, β3[ (]p8, p4[, ]l2eq4, l2eq3[), (]p4, p5[, ]l2eq4, l2eq3[), (]p5, p7[, ]l2eq4, l2eq2[)
[β3, β4] (]p8, p5[, ]l2eq4, l2eq3[), (]p5, p7[, ]l2eq4, l2eq2[)
Table 4 Cells where Eq. (13) does not have any real root with a1∈]a11, a12[ and a2∈]a21, a22[
[β1, β2[ (]p8, p3[, ]l2eq2, l2eq5[), (]p3, p4[, ]l2eq2, l2eq5[), (]p4, p5[, ]l2eq1, l2eq5[), (]p5, p7[, ]l2eq1, l2eq5[)
[β2, β3[ (]p8, p4[, ]l2eq2, l2eq5[), (]p4, p5[, ]l2eq1, l2eq5[), (]p5, p7[, ]l2eq1, l2eq5[)
[β3, β4] (]p8, p5[, ]l2eq1, l2eq5[), (]p5, p7[, ]l2eq1, l2eq5[)
l2eq0= 0 l2eq1= 0 l2eq2= 0 l2eq3= 0 a21= a1 p8= 0 p8= 0 p4= 0 (a) (b)
Fig. 5 Cells where the IRSBot-2 cannot reach any constraint singularity for: (a) a1= 1, β =
arcsin(1/√3) and l2eq< (a1− a2) sin β ; (b) a1= 1, β = arcsin(1/
√
3) and l2eq> (a1− a2) sin β
Figure 5(a) (Fig. 5(b), resp.) illustrates the cells where Eq. (12) (Eq. (13), resp.) does not have any real root, namely, the sets of design parameters that prevent the IRSBot-2 from reaching any constraint singularity for a1= 1, β = arcsin(1/√3) and
8 Coralie Germain, S´ebastien Briot, St´ephane Caro and Philippe Wenger
of constraint singularity-free designs is higher with l2eq> (a1− a2) sin β than with
l2eq< (a1− a2) sin β .
5 Conclusions
This paper dealt with the constraint analysis of the IRSBot-2 throughout its param-eter space. Its constraint singularities were analyzed in its paramparam-eter space with a method based on the notion of Discriminant Varieties and Cylindrical Algebraic Decomposition. This method allowed us to convert a kinematic problem into an al-gebraic one. Then, a deep analysis was carried out in order to determine the sets of design parameters of the distal modules that prevent the IRSBot-2 from reach-ing any constraint sreach-ingularity. To the best of our knowledge, such an analysis had never been performed before. The design parameters associated with the proximal modules for the IRSBot-2 to be assembled will be determined in a future work.
6 Acknowledgment
This work was conducted with the support of the French National Research Agency (Project ANR-2011-BS3-006-01-ARROW). The authors also thank Damien Chab-lat for his great help with the Siropa Maple library.
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